CBSE Class 12 Mathematics 2024 question paper (65/5)

Maximum marks 80 · Time 3 hours · 3 sets

Try each question first, then open its model answer. Where the paper offers a choice, both questions are shown with OR between them.

Section A

1 mark each

Q.11 mark $f : \mathbb{R} \to \mathbb{R}$
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Q.21 mark $A = \begin{bmatrix} a & c & -1 \\ b & 0 & 5 \\ 1 & -5 & 0 \end{bmatrix}$
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Q.31 mark $\left|\mathrm{adj}{\cdot}\mathrm{A}\right| = 8$
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Q.41 mark $\begin{bmatrix} 7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix}$
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Q.51 mark $\begin{bmatrix} x & 2 & 0 \end{bmatrix} \begin{bmatrix} 5 \\ -1 \\ x \end{bmatrix} = \begin{bmatrix} 3 & 1 \end{bmatrix} \begin{bmatrix} -2 \\ x \end{bmatrix}$
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Q.61 mark $A^2$
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Q.71 mark $xe^y = 1$
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Q.81 mark $e^{\sin^2 x}$
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Q.91 mark $f(x) = \dfrac{x}{2} + \dfrac{2}{x}$
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Q.101 mark $y = 7x - x^3$
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Q.111 mark $\displaystyle\int \dfrac{1}{x(\log x)^2}\, dx$
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Q.121 mark $\displaystyle\int_{-1}^{1} x\,|x|\, dx$
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Q.131 mark $y^2 = 4x$
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Q.141 mark $\dfrac{d^4y}{dx^4} - \sin\left(\dfrac{d^2y}{dx^2}\right) = 5$
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Q.151 mark $\vec{p}$
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Q.161 mark $\dfrac{x}{1} = \dfrac{y}{-1} = \dfrac{z}{0}$
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Q.171 mark $(1, -3, 2)$
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Q.181 mark $P(A/B) = P(B/A) \ne 0$
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Direction : In questions numbers 19 and 20, two statements are given one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the following options :
Q.191 mark $y = \cos^{-1}(x)$
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Q.201 mark $\vec{a} = 6\hat{i} + 2\hat{j} - 8\hat{k}$
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Section B

2 marks each

Q.212 marks $\sin^{-1}\left[\mathrm{k}\tan\left(2\cos^{-1}\dfrac{\sqrt{3}}{2}\right)\right] = \dfrac{\pi}{3}$
Q.22 (a)2 marks $\mathrm{f}(x) = \begin{cases} x\sin\left(\dfrac{1}{x}\right), & x \ne 0 \\ 0, & x = 0 \end{cases}$
OR
Q.22 (b)2 marks $\mathrm{f}(x) = |x - 5|$
Q.232 marks $^2$
Q.24 (a)2 marks $\displaystyle\int \cos^3 x\; \mathrm{e}^{\log \sin x}\, \mathrm{d}x$
OR
Q.24 (b)2 marks $\displaystyle\int \dfrac{1}{5 + 4x - x^2}\, \mathrm{d}x$
Q.252 marks $(2, 3, -5)$

Section C

3 marks each

Q.26 (a)3 marks $\dfrac{\mathrm{d}y}{\mathrm{d}x}$
OR
Q.26 (b)3 marks $\sqrt{1 - x^2} + \sqrt{1 - \mathrm{y}^2} = \mathrm{a}(x - \mathrm{y})$
Q.273 marks $x = \mathrm{a} \sin^3 \theta$
Q.28 (a)3 marks $\displaystyle\int_{0}^{\pi} \dfrac{\mathrm{e}^{\cos x}}{\mathrm{e}^{\cos x} + \mathrm{e}^{-\cos x}}\, \mathrm{d}x$
OR
Q.28 (b)3 marks $\displaystyle\int \dfrac{2x + 1}{(x + 1)^2\,(x - 1)}\, \mathrm{d}x$
Q.29 (a)3 marks $\dfrac{\mathrm{d}y}{\mathrm{d}x} - 2x\mathrm{y} = 3x^2\, \mathrm{e}^{x^2}$
OR
Q.29 (b)3 marks $x^2\, \mathrm{d}y + \mathrm{y}(x + \mathrm{y})\, \mathrm{d}x = 0$
Q.303 marks $2\hat{i} - \hat{j} + \hat{k}$
Q.313 marks $P(X \ge 3)$

X	1	2	3	4	5
P(X)	0.2	a	a	0.2	b

Section D

5 marks each

Q.32 (a)5 marks $\mathrm{A} = \begin{bmatrix} 1 & 2 & -3 \\ 2 & 0 & -3 \\ 1 & 2 & 0 \end{bmatrix}$
OR
Q.32 (b)5 marks $\begin{bmatrix} 1 & 2 & -3 \\ 2 & 3 & 2 \\ 3 & -3 & -4 \end{bmatrix}$
Q.335 marks $4x^2 + \mathrm{y}^2 = 36$
Q.34 (a)5 marks $(2, 3, -8)$
OR
Q.34 (b)5 marks $\mathrm{L}_1$
Q.355 marks $\mathrm{Z} = 60x + 40\mathrm{y}$

Section E

Students of a school are taken to a railway museum to learn about railways heritage and its history. An exhibit in the museum depicted many rail lines on the track near the railway station. Let L be the set of all rail lines on the railway track and R be the relation on L defined by R = $\{(l_1, l_2) : l_1$ is parallel to $l_2\}$ On the basis of the above information, answer the following questions :
Q.36 (a) (i)1 mark $Find whether the relation R is symmetric or not.$
Q.36 (a) (ii)1 mark $Find whether the relation R is transitive or not.$
Q.36 (a) (iii)2 marks $\mathrm{y} = 3x + 2$
OR
Q.36 (b)4 marks $\{(l_1, l_2) : l_1$
A rectangular visiting card is to contain 24 sq.cm. of printed matter. The margins at the top and bottom of the card are to be 1 cm and the margins on the left and right are to be 1½ cm as shown below : On the basis of the above information, answer the following questions :
Q.37 (i)2 marks $x$
Q.37 (ii)2 marks $Obtain the dimensions of the card of minimum area.$
A departmental store sends bills to charge its customers once a month. Past experience shows that 70% of its customers pay their first month bill in time. The store also found that the customer who pays the bill in time has the probability of 0.8 of paying in time next month and the customer who doesn't pay in time has the probability of 0.4 of paying in time the next month. Based on the above information, answer the following questions :
Q.38 (i)1 mark $\mathrm{E}_1$
Q.38 (ii)1 mark $P(\mathrm{A}|\mathrm{E}_1)$
Q.38 (iii) (a)2 marks $Find the probability of customer paying second month's bill in time.$
OR
Q.38 (iii) (b)2 marks $Find the probability of customer paying first month's bill in time if it is found that customer has paid the second month's bill in time.$

Section A

1 mark each

Q.11 mark $e^{\sin^2 x}$
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Q.21 mark $\left|\mathrm{A}\right| = -2$
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Q.31 mark $f(x) = \dfrac{x}{2} + \dfrac{2}{x}$
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Q.41 mark $y = 7x - x^3$
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Q.51 mark $3 \times 2$
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Q.61 mark $f : \mathbb{R} \to \mathbb{R}$
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Q.71 mark $\sin(xy) = 1$
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Q.81 mark $\begin{bmatrix} 7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix}$
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Q.91 mark $A^2$
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Q.101 mark $\left|\mathrm{adj}{\cdot}\mathrm{A}\right| = 8$
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Q.111 mark $\displaystyle\int_{\pi/4}^{\pi/2} \cot\theta \; \mathrm{cosec}^2\theta \; d\theta$
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Q.121 mark $\displaystyle\int \dfrac{dx}{\sqrt{9 - 4x^2}}$
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Q.131 mark $y^2 = 4x$
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Q.141 mark $\dfrac{dy}{dx} = e^{x + y}$
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Q.151 mark $\dfrac{x}{1} = \dfrac{y}{-1} = \dfrac{z}{0}$
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Q.161 mark $(1, -3, 2)$
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Q.171 mark $P(A/B) = P(B/A) \ne 0$
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Q.181 mark $\vec{p}$
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Direction : In questions numbers 19 and 20, two statements are given one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the following options :
Q.191 mark $\vec{a} = 6\hat{i} + 2\hat{j} - 8\hat{k}$
Tap an option to check your answer.
Q.201 mark $y = \cos^{-1}(x)$
Tap an option to check your answer.

Section B

2 marks each

Q.212 marks $= \sin^{-1}\left(\dfrac{\sqrt{2}}{2}\right) + \cos^{-1}\left(-\dfrac{1}{2}\right)$
Q.22 (a)2 marks $\displaystyle\int \cos^3 x\; \mathrm{e}^{\log \sin x}\, \mathrm{d}x$
OR
Q.22 (b)2 marks $\displaystyle\int \dfrac{1}{5 + 4x - x^2}\, \mathrm{d}x$
Q.232 marks $^3$
Q.242 marks $(2, 3, -5)$
Q.25 (a)2 marks $\mathrm{f}(x) = \begin{cases} x\sin\left(\dfrac{1}{x}\right), & x \ne 0 \\ 0, & x = 0 \end{cases}$
OR
Q.25 (b)2 marks $\mathrm{f}(x) = |x - 5|$

Section C

3 marks each

Q.26 (a)3 marks $\dfrac{\mathrm{d}y}{\mathrm{d}x} - 2x\mathrm{y} = 3x^2\, \mathrm{e}^{x^2}$
OR
Q.26 (b)3 marks $x^2\, \mathrm{d}y + \mathrm{y}(x + \mathrm{y})\, \mathrm{d}x = 0$
Q.273 marks $x$
Q.28 (a)3 marks $\dfrac{\mathrm{d}y}{\mathrm{d}x}$
OR
Q.28 (b)3 marks $\sqrt{1 - x^2} + \sqrt{1 - \mathrm{y}^2} = \mathrm{a}(x - \mathrm{y})$
Q.29 (a)3 marks $\displaystyle\int_{0}^{\pi} \dfrac{\mathrm{e}^{\cos x}}{\mathrm{e}^{\cos x} + \mathrm{e}^{-\cos x}}\, \mathrm{d}x$
OR
Q.29 (b)3 marks $\displaystyle\int \dfrac{2x + 1}{(x + 1)^2\,(x - 1)}\, \mathrm{d}x$
Q.303 marks $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$
Q.313 marks $Bag I contains 3 red and 4 black balls, Bag II contains 5 red and 2 black balls. Two balls are transferred at random from Bag I to Bag II and then a ball is drawn at random from Bag II. Find the probability that the drawn ball is red in colour.$

Section D

5 marks each

Q.32 (a)5 marks $(2, 3, -8)$
OR
Q.32 (b)5 marks $\mathrm{L}_1$
Q.33 (a)5 marks $\mathrm{A} = \begin{bmatrix} 1 & 2 & -3 \\ 2 & 0 & -3 \\ 1 & 2 & 0 \end{bmatrix}$
OR
Q.33 (b)5 marks $\begin{bmatrix} 1 & 2 & -3 \\ 2 & 3 & 2 \\ 3 & -3 & -4 \end{bmatrix}$
Q.345 marks $4x^2 + \mathrm{y}^2 = 36$
Q.355 marks $\mathrm{Z} = 300x + 600\mathrm{y}$

Section E

A departmental store sends bills to charge its customers once a month. Past experience shows that 70% of its customers pay their first month bill in time. The store also found that the customer who pays the bill in time has the probability of 0.8 of paying in time next month and the customer who doesn't pay in time has the probability of 0.4 of paying in time the next month. Based on the above information, answer the following questions :
Q.36 (i)1 mark $\mathrm{E}_1$
Q.36 (ii)1 mark $P(\mathrm{A}|\mathrm{E}_1)$
Q.36 (iii) (a)2 marks $Find the probability of customer paying second month's bill in time.$
OR
Q.36 (iii) (b)2 marks $Find the probability of customer paying first month's bill in time if it is found that customer has paid the second month's bill in time.$
Students of a school are taken to a railway museum to learn about railways heritage and its history. An exhibit in the museum depicted many rail lines on the track near the railway station. Let L be the set of all rail lines on the railway track and R be the relation on L defined by R = $\{(l_1, l_2) : l_1$ is parallel to $l_2\}$ On the basis of the above information, answer the following questions :
Q.37 (a) (i)1 mark $Find whether the relation R is symmetric or not.$
Q.37 (a) (ii)1 mark $Find whether the relation R is transitive or not.$
Q.37 (a) (iii)2 marks $\mathrm{y} = 3x + 2$
OR
Q.37 (b)4 marks $\{(l_1, l_2) : l_1$
A rectangular visiting card is to contain 24 sq.cm. of printed matter. The margins at the top and bottom of the card are to be 1 cm and the margins on the left and right are to be 1½ cm as shown below : On the basis of the above information, answer the following questions :
Q.38 (i)2 marks $x$
Q.38 (ii)2 marks $Obtain the dimensions of the card of minimum area.$

Section A

1 mark each

Q.11 mark $\vec{p}$
Tap an option to check your answer.
Q.21 mark $\mathrm{A} = \begin{bmatrix} 2 & -1 & 1 \\ \lambda & 2 & 0 \\ 1 & -2 & 3 \end{bmatrix}$
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Q.31 mark $\dfrac{x}{1} = \dfrac{y}{-1} = \dfrac{z}{0}$
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Q.41 mark $(1, -3, 2)$
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Q.51 mark $\mathrm{A} = \begin{bmatrix} x & 0 \\ 1 & 1 \end{bmatrix}$
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Q.61 mark $y = 7x - x^3$
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Q.71 mark $\mathrm{f}(x) = \begin{vmatrix} x^2 & \sin x \\ \mathrm{p} & -1 \end{vmatrix}$
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Q.81 mark $P(A/B) = P(B/A) \ne 0$
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Q.91 mark $f : \mathbb{R} \to \mathbb{R}$
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Q.101 mark $\left|\mathrm{adj}{\cdot}\mathrm{A}\right| = 8$
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Q.111 mark $\displaystyle\int_{-2}^{3} x^2\, \mathrm{d}x = \mathrm{k}\int_{0}^{2} x^2\, \mathrm{d}x + \int_{2}^{3} x^2\, \mathrm{d}x$
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Q.121 mark $\displaystyle\int_{1}^{e} \log x\, \mathrm{d}x$
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Q.131 mark $y = \sqrt{x}$
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Q.141 mark $\dfrac{\mathrm{d}^3\mathrm{y}}{\mathrm{d}x^3} + x\left(\dfrac{\mathrm{d}\mathrm{y}}{\mathrm{d}x}\right)^5 = 4\log\left(\dfrac{\mathrm{d}^4\mathrm{y}}{\mathrm{d}x^4}\right)$
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Q.151 mark $\begin{bmatrix} 7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix}$
Tap an option to check your answer.
Q.161 mark $A^2$
Tap an option to check your answer.
Q.171 mark $e^{\sin^2 x}$
Tap an option to check your answer.
Q.181 mark $f(x) = \dfrac{x}{2} + \dfrac{2}{x}$
Tap an option to check your answer.
Direction : In questions numbers 19 and 20, two statements are given one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the following options :
Q.191 mark $y = \cos^{-1}(x)$
Tap an option to check your answer.
Q.201 mark $\vec{a} = 6\hat{i} + 2\hat{j} - 8\hat{k}$
Tap an option to check your answer.

Section B

2 marks each

Q.212 marks $\cos^{-1}x + \cos^{-1}\left[\dfrac{x}{2} + \dfrac{\sqrt{3 - 3x^2}}{2}\right]$
Q.22 (a)2 marks $\displaystyle\int \cos^3 x\; \mathrm{e}^{\log \sin x}\, \mathrm{d}x$
OR
Q.22 (b)2 marks $\displaystyle\int \dfrac{1}{5 + 4x - x^2}\, \mathrm{d}x$
Q.232 marks $^2$
Q.242 marks $(2, 3, -5)$
Q.25 (a)2 marks $\mathrm{f}(x) = \begin{cases} x\sin\left(\dfrac{1}{x}\right), & x \ne 0 \\ 0, & x = 0 \end{cases}$
OR
Q.25 (b)2 marks $\mathrm{f}(x) = |x - 5|$

Section C

3 marks each

Q.26 (a)3 marks $\displaystyle\int_{0}^{\pi} \dfrac{\mathrm{e}^{\cos x}}{\mathrm{e}^{\cos x} + \mathrm{e}^{-\cos x}}\, \mathrm{d}x$
OR
Q.26 (b)3 marks $\displaystyle\int \dfrac{2x + 1}{(x + 1)^2\,(x - 1)}\, \mathrm{d}x$
Q.273 marks $\mathrm{y} = (\tan^{-1}x)^2$
Q.28 (a)3 marks $\dfrac{\mathrm{d}y}{\mathrm{d}x} - 2x\mathrm{y} = 3x^2\, \mathrm{e}^{x^2}$
OR
Q.28 (b)3 marks $x^2\, \mathrm{d}y + \mathrm{y}(x + \mathrm{y})\, \mathrm{d}x = 0$
Q.29 (a)3 marks $\dfrac{\mathrm{d}y}{\mathrm{d}x}$
OR
Q.29 (b)3 marks $\sqrt{1 - x^2} + \sqrt{1 - \mathrm{y}^2} = \mathrm{a}(x - \mathrm{y})$
Q.303 marks $(\vec{b} + \vec{c})$
Q.313 marks $An urn contains 3 red and 2 white marbles. Two marbles are drawn one by one with replacement from the urn. Find the probability distribution of the number of white balls. Also, find the mean of the number of white balls drawn.$

Section D

5 marks each

Q.325 marks $4x^2 + \mathrm{y}^2 = 36$
Q.33 (a)5 marks $(2, 3, -8)$
OR
Q.33 (b)5 marks $\mathrm{L}_1$
Q.34 (a)5 marks $\mathrm{A} = \begin{bmatrix} 1 & 2 & -3 \\ 2 & 0 & -3 \\ 1 & 2 & 0 \end{bmatrix}$
OR
Q.34 (b)5 marks $\begin{bmatrix} 1 & 2 & -3 \\ 2 & 3 & 2 \\ 3 & -3 & -4 \end{bmatrix}$
Q.355 marks $\mathrm{Z} = 6x + 3\mathrm{y}$

Section E

A rectangular visiting card is to contain 24 sq.cm. of printed matter. The margins at the top and bottom of the card are to be 1 cm and the margins on the left and right are to be 1½ cm as shown below : On the basis of the above information, answer the following questions :
Q.36 (i)2 marks $x$
Q.36 (ii)2 marks $Obtain the dimensions of the card of minimum area.$
A departmental store sends bills to charge its customers once a month. Past experience shows that 70% of its customers pay their first month bill in time. The store also found that the customer who pays the bill in time has the probability of 0.8 of paying in time next month and the customer who doesn't pay in time has the probability of 0.4 of paying in time the next month. Based on the above information, answer the following questions :
Q.37 (i)1 mark $\mathrm{E}_1$
Q.37 (ii)1 mark $P(\mathrm{A}|\mathrm{E}_1)$
Q.37 (iii) (a)2 marks $Find the probability of customer paying second month's bill in time.$
OR
Q.37 (iii) (b)2 marks $Find the probability of customer paying first month's bill in time if it is found that customer has paid the second month's bill in time.$
Students of a school are taken to a railway museum to learn about railways heritage and its history. An exhibit in the museum depicted many rail lines on the track near the railway station. Let L be the set of all rail lines on the railway track and R be the relation on L defined by R = $\{(l_1, l_2) : l_1$ is parallel to $l_2\}$ On the basis of the above information, answer the following questions :
Q.38 (a) (i)1 mark $Find whether the relation R is symmetric or not.$
Q.38 (a) (ii)1 mark $Find whether the relation R is transitive or not.$
Q.38 (a) (iii)2 marks $\mathrm{y} = 3x + 2$
OR
Q.38 (b)4 marks $\{(l_1, l_2) : l_1$